Help with derivative questions in beginning calc?

Question

I am getting the right answer, but im not doing these right......so far we have learned the product, quotient, exponential and chain rule (as well as the basics)
please show work and explain if its not evident.....sorry there are 4

find the derivative of cot^3(squareroot of x

derivitive of y         x/(x+y)

d^2y/dx^2 for y=(1-x)/(2-x)

differentiate and simplify y=sin^2(x)-cos^2(x)

norcekri · Accepted Answer

I'll shoot at the last one:

y=sin(x)sin(x) - cos(x)cos(x)
y' = [sin(x)cos(x) + cos(x)sin(x)] - [-sin(x)cos(x) - cos(x)sin(x)]
y' = 2sin(x)cos(x) + 2 sin(x)cos(x) = 4sin(x)cos(x)

If I recall correctly, this is 2 sin(2x), no?  Given the trig identity on the original equation, this looks correct.

For the first one, I'd re-cast it as x^(1/2) * sin^(-3)(x).  Use the product rule on the whole, the exponential rule on each.

I'm not sure what's going on in the second one; do you have to solve for y first?  It looks like there might be a missing equals sign.

For the third, first multiply through by -1 to get (x-1)/(x-2).  This will simplify the quotient rule -- important, since you're using it twice.  Let's see ... for dy/dx ...

[(x-1)1 - (x-2)1] / (x-2)^2 
1 / (x-2)^2
(x-2)^(-2)
You should be able to do the second derivative from the first now.

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Help with derivative questions in beginning calc?

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